# Derive the Formula for the Rotational Energy of a Diatomic Molecule

Here’s an example that involves finding the rotational energy spectrum of a diatomic molecule. The figure shows the setup: A rotating diatomic molecule is composed of two atoms with masses *m*_{1} and *m*_{2}. The first atom rotates at *r* = *r*_{1}, and the second atom rotates at *r* = *r*_{2}. What’s the molecule’s rotational energy?

A rotating diatomic molecule.

The Hamiltonian is

I is the rotational moment of inertia, which is

where *r* = |*r*_{1} – *r*_{2}| and

Because

Therefore, the Hamiltonian becomes

So applying the Hamiltonian to the eigenstates, | *l*, *m** *>, gives you the following:

And as you know,

so this equation becomes

And because H | *l*, *m* > = E | *l*, *m* >, you can see that

And that’s the energy as a function of *l*, the angular momentum quantum number.